On the other hand, an important class of solutions is constructed by a finite-time discretization and Galerkin approximation scheme, see [1]}, which is needed when dealing with mixed Neumann-Dirichlet boundary conditions on Lipschitz domains.
The well-posedness proof in [2]} is based on these methods.
For this class of solutions it is not immediately clear that the approximation assumption holds, since it does not follow from the existence proof presented in [1]} in a natural manner.
It can be shown that there exists a sequence of approximate continuous solutions, which would be sufficient for our purposes.
Nevertheless, this approach relies on the availability of the well-posedness of an appropriate initial- / boundary value problem rather than invoking only the definition of local solutions.
